geometric time delay interferometry

PHYSICAL REVIEW D 72, 042003 (2005)

Geometric time delay interferometry

Michele VallisneriJet Propulsion Laboratory, California Institute of Technology, Pasadena, California 91109, USA

(Received 4 May 2005; published 12 August 2005)

1550-7998=20

The space-based gravitational-wave observatory LISA, a NASA-ESA mission to be launched after2012, will achieve its optimal sensitivity using time delay interferometry (TDI), a LISA-specific techniqueneeded to cancel the otherwise overwhelming laser noise in the interspacecraft phase measurements.The TDI observables of the Michelson and Sagnac types have been interpreted physically as the virtualmeasurements of a synthesized interferometer. In this paper, I present Geometric TDI, a new and intuitiveapproach to extend this interpretation to all TDI observables. Unlike the standard algebraic formalism,Geometric TDI provides a combinatorial algorithm to explore exhaustively the space of second-generation TDI observables (i.e., those that cancel laser noise in LISA-like interferometers withtime-dependent arm lengths). Using this algorithm, I survey the space of second-generation TDIobservables of length (i.e., number of component phase measurements) up to 24, and I identify alternative,improved forms of the standard second-generation TDI observables. The alternative forms haveimproved high-frequency gravitational-wave sensitivity in realistic noise conditions (because they havefewer nulls in the gravitational-wave and noise response functions), and are less susceptible toinstrumental gaps and glitches (because their component phase measurements span shorter timeperiods).

DOI: 10.1103/PhysRevD.72.042003 PACS numbers: 04.80.Nn, 07.60.Ly, 95.55.Ym

1A variant of the technique uses combinations of one-way andtwo-way phase measurements, generated by locking five of thesix LISA lasers to the last one, as described by Tinto andcolleagues [12]. In this paper we shall consider only the one-way formalism, but our results could be applied with superficialmodifications also to the two-way variant.

I. INTRODUCTION

The Laser Interferometer Space Antenna (LISA) is ajoint NASA-ESA deep-space mission to be launched after2012, aimed at detecting and studying gravitational waves(GWs) with frequencies between 10�5 and 10�1 Hz [1].LISA will provide access to GW sources that areoutside the reach of ground-based interferometric GWdetectors [2], such as the binaries of compact stellar objectsin our galaxy, the mergers of massive and supermassiveblack holes, and the gravitational captures of compactobjects by the supermassive black holes at the center ofgalaxies [3].

LISA consists of three widely separated spacecraft,flying around the Sun in a quasiequilateral triangular con-figuration and exchanging phase-coherent laser signals.LISA relies on picometer interferometry to measureGWs as modulations in the distance between the space-craft. The greatest challenge to achieving this measure-ment is the phase noise of the LISA lasers, which is largerthan the GW-induced response by many orders of magni-tude, and which cannot be removed by conventional phase-matching interferometry because the LISA arm lengths aregrossly unequal and changing continuously. Time DelayInterferometry (TDI), developed by J. W. Armstrong,F. B. Estabrook, M. Tinto, and others [4–11], is theLISA-specific technique that will be used to combinethe laser-noise-laden one-way phase measurements per-formed between the three spacecraft1 into virtual interfero-metric observables where laser noise is reduced by severalorders of magnitude.

05=72(4)=042003(18)$23.00 042003

TDI was initially developed using ad hoc algebraicreasoning for the case of a stationary LISA configurationwith unequal but constant arm lengths (first-generationTDI, see [4,5]). It was later modified to work also in thecase of a rotating LISA constellation (modified TDI, see[8–11]) and of linearly changing arm lengths (second-generation TDI, see [10,11]). First-generation and modi-fied TDI were given a rigorous mathematical foundation inthe theory of algebraic syzygies on moduli [7], providingtools to generate all possible TDI observables and todetermine which observables are optimally sensitive toGWs [13]. Unfortunately, this algebraic treatment cannotbe extended easily to second-generation TDI, which is theversion that must be used in practice.

In this paper, I give a new derivation of first-generation,modified, and second-generation TDI using a geometricapproach that emphasizes the physical interpretation ofTDI observables as synthesized interferometric measure-ments [4,10,14], extending it to all known observables.What is more, this geometric approach to TDI (in short,Geometric TDI) allows the exhaustive enumeration of allTDI observables of any length, and it leads to alternative,improved forms of the standard TDI observables, charac-

-1 2005 The American Physical Society

http://dx.doi.org/10.1103/PhysRevD.72.042003

FIG. 1 (color online). In this idealization of the basic time-

MICHELE VALLISNERI PHYSICAL REVIEW D 72, 042003 (2005)

terized by better GW sensitivity at high frequencies inrealistic noise conditions, by lesser demands on the mea-surement system, and by reduced susceptibility to gaps andglitches.

More specifically, all TDI observables display nulls intheir noise and GW responses at frequency multiples of theinverse arm-crossing light times.2 Because these zerosoccur at the same frequencies and with the same ordersfor noise and GWs, the ideal GW sensitivity after success-ful laser-noise suppression is finite and comparable to thesensitivity at nearby frequencies. The actual sensitivity,however, is likely to be degraded, either because noiseleaks into the nulls from the sides [15], or because themeasurement system has insufficient dynamical range toresolve the tiny signals within the nulls. This problem ismitigated with the alternative observables, which have halfas many response-function nulls as the standard forms.

In addition, because the alternative observables are, as itwere, folded versions of their standard forms, they have asmaller temporal footprint: that is, they are written as sumsof one-way phase measurements that span a shorter timeperiod. This property can be advantageous in the presenceof instrumental gaps or glitches, which would then con-taminate a smaller portion of the data set; a reducedtemporal footprint means also that a shorter continuousset of phase data needs to be collected before TDI observ-ables can begin to be assembled.

This paper is organized as follows: Section II describesGeometric TDI: in Sec. II A, I introduce the basic GW-sensitive phase measurement; in Sec. II B, I discuss itsintegration into laser-noise-canceling observables accord-ing to the Geometric TDI principle; in Secs. II C and II D , Igive a new derivation of the observables of first-generation,modified, and second-generation TDI, and I interpret themgeometrically; in Sec. II E, I show how to enumerate ex-haustively all possible observables by representing them aslink strings; last, in Sec. II F I extend our formalism,developed for simplicity by considering only three inde-pendent LISA lasers, to the realistic case of six lasers.Section III reports on the exhaustive survey of allsecond-generation TDI observables consisting of up to 24separate phase measurements: in Secs. III A and III B , Idiscuss the alternative forms of the standard second-generation TDI observables, and present their practicaladvantages for the implementation of TDI; in Sec. III C, Idescribe the previously unknown second-generation TDIobservables of length 18 and more. Lastly, in Sec. IV, Ipresent my conclusions. The appendices contain rules andproofs omitted from the main text, and explicit algebraicexpression for the second-generation TDI observables oflength 16.

2The responses are exactly null only in the limit of equal LISAarm lengths. For realistic, time-evolving LISA geometries thenulls are spread into narrow dips; however, these are deepenough that the qualitative discussion to follow still applies.

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As customary, I set G � c � 1 except where specifiedotherwise.

II. A GEOMETRIC VIEW OF TIME DELAYINTERFEROMETRY

How is LISA an interferometer other than by name? Theloosest dictionary definition of ‘‘interferometer’’ (some-thing like ‘‘a device that combines the signals radiatingfrom a common source and received at different locations,or at the same location after traveling different paths’’)does not seem to apply to LISA, whose TDI GW observ-ables are combinations of the phase-difference measure-ments between as many as six laser sources. In fact,interferometry is not needed, strictly speaking, to measureGWs, but only to remove the otherwise deafening phasenoise produced by the LISA lasers. The basic principle ofGW measurement employed by LISA is noninterferomet-ric, as we can see from the idealized experimental setup ofa time-transport link between two ideal clocks (see Fig. 1).

A. The basic time-transport observable

Consider a plane GW propagating across the Minkowskibackground geometry and written in the transverse-traceless (TT) gauge [16]. The wave is traveling alongthe x direction and has ‘‘�’’ polarization along the y andz directions. We can then write the spacetime metric as�� hTT�, where

transport observable used with LISA, two ideal clocks travelalong geodesics, with clock 1 continuously transferring itsproper time to clock 2 by way of pulsed light signals. GWsare measured as the fluctuations in the time of flight between theclocks (see main text).

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3See the TDI Rosetta Stone, first printed in Ref. [20] andupdated at www.vallis.org/tdi, for a mapping between the differ-ent conventions used in the literature.

GEOMETRIC TIME DELAY INTERFEROMETRY PHYSICAL REVIEW D 72, 042003 (2005)

hTT� � h��t� x��ezz � eyy�: (1)

Consider also two ideal clocks, 1 and 2, marking theirproper times t1 and t2, and sitting at constant spatialcoordinates ~p1 � f0; 0; 0g and ~p2 � f0; 0; Lg in the TTframe. In this gauge, constant-coordinate world lines aregeodesics, so the effect of the GWs is not to exert forces (asit were) on test particles, but to modulate the distancebetween them. By way of light signals, clock 1 is contin-uously sending its time t1 to clock 2, where t1 is comparedwith the local time t2, yielding the difference

�t12 � t2�t� � t1�t� L12�t�� L12�t�: (2)

Here t is the TT coordinate time and L12�t� is the time offlight between the two clocks, as experienced by the laserpulse that arrives at clock 2 at time t. We are assuming thatthe two clocks have been synchronized so that in theabsence of GWs they both mark the coordinate time t. Tofirst-order in h�, L12�t� is

L12�t� � L�1

2

Z t

t�Lh��t�dt; (3)

the x coordinate dependence of the GW does not appear inEq. (3) because the two clocks sit on the same constant-xwave fronts. If the rates of the two clocks remain synchro-nized (dt1=dt � dt2=dt), then the time derivative d�t12=dtis directly proportional to difference of the GW strains atthe events of pulse reception and emission,

d�t12dt

�d�t12dt2

�1

2�h��t� � h��t� L��: (4)

This is our GW observable. In the Fourier domain, the(power) response function of d�t12=dt to GWs is j1�exp��2�ifL�j2=4 � sin2��fL�. Thus d�t12=dt is insen-sitive to GWs of frequencies f� 1=L or f ’ k=L (withinteger k). Expression (4) is the basic building block usedto derive the LISA response to GWs, as well as the Dopplerresponse used in spacecraft-tracking GW searches [17] andthe timing-residual response used in pulsar-timing searches[18].

To relate this idealized experimental setup to LISA, wereplace the ideal clocks with the LISA lasers, and obtainproper time by dividing the lasers’ phase by their fre-quency. Each of the three LISA spacecraft contains twooptical benches oriented facing the other two spacecraft;on each bench, the appropriately named phasemeters com-pare the phase of the incoming lasers against the localreference laser. As written, Eq. (4) involves a comparisonof laser frequencies: we choose to develop our argumentsin terms of these, since it is more convenient to deal withinstrumental responses that are directly proportional to thephysical observable of interest (the GW strain) rather thanto its time integral. Generalizing Eq. (4) to arbitrary plane-GW and spacecraft geometries, and adopting a LISA-specific language, we come to the Estabrook-Wahlquist

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two-pulse response [19]

y12�t� �1

2

ni12�t�nj12�t��h

TTij �p

s2�t�; t� � h

TTij �p

s1�ts�; ts��

1� nm12�t�km;

(5)

with this indexing, Eq. (5) describes the frequency-difference measurement performed on spacecraft 2 to com-pare the local laser to the laser incoming from spacecraft 1.In this equation:

(i) k

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j is the spatial propagation vector of the planeGW;

(ii) p
s1�t� and ps2�t� are the spatial TT coordinates of thetwo spacecraft;
(iii) t
is the time of pulse reception, and therefore ofmeasurement;
(iv) t
s is the time of pulse emission, as determinedimplicitly by jps2�t� � p
s1�ts�j � t� ts;

(v) n
m12�t� is the unit vector along the trajectory of thelight pulse (labeled by the time of reception t),given by nm12�t� � �p
s2�t� � p

s1�ts��=jp

s2�t� �

ps1�ts�j.
Equation (5) is known as the two-pulse response becausean impulsive GW is registered twice in each yij�t� observ-able, once when it impinges on the emitting spacecraft i,and once, a time L later, when it impinges on the receivingspacecraft j.
In the literature on TDI, it is customary to label the LISAarms by the index of the opposite spacecraft. We shall do soin this paper, using primed and unprimed indices to denotethe oriented LISA arms (with orientation following thedirection of laser transmission) according to the conven-tion3 f1; 2; 3g f3! 2; 1! 3; 2! 1g and f10; 20; 30g f2! 3; 3! 1; 1! 2g. We shall then denote by Ll thepropagation time experienced by a laser pulse travelingalong the oriented arm l. We shall also find it useful, attimes, to augment the yij phase-measurement notation witha middle index, corresponding to the oriented arm trav-ersed by the laser pulse being measured. (In fact, theprimed or unprimed middle index would be sufficient toidentify the phase measurement completely, and we shallexploit this property in Sec. II E when we represent TDIobservables as link strings.) See Fig. 2 for an example ofthis convention at work.

B. The Geometric TDI principle

Unfortunately, GWs cannot be read off directly from theyij measurements, because the fluctuations Ci�t� of thelaser frequencies (i.e., the laser phase noises) come intothe yij�t� as

FIG. 3. Left.—The arrows of this closed loop reproduce thepaths of light in a standard equal-arm Michelson interferometer,and the corresponding time-ordered sum of phase measurements[Eq. (7)] reproduces the phase-difference output of the interfer-ometer. Right.—For unequal arm lengths, laser phase noisecancellation can be recovered by having both interfering beamstravel along each arm once, building up the same light-traveltime. Compare with Fig. 1 of Ref. [10].

FIG. 2. These time delayed sums and differences of two yijmeasurements cancel laser phase noise at time t. In all of them,two laser pulses arrive at, or depart from, the same spacecraft attime t.


yij�t� � Ci�ts� � Cj�t� � GWs; (6)

the LISA lasers have Ci�t� of (single-sided, square-root)spectral density�30Hz=

��Hzp

, several orders of magnitudestronger than the weakest GWs detectable by LISA, whichare at the level of the other two fundamental LISA noises(known together as secondary noises): the shot noise at thephasemeter, as determined by the power of the lasers andby the distance between the spacecraft, and the accelera-tion noise of the proof masses enclosed within each opticalbench, which are used to reference the frequency measure-ments to freely falling world lines. [Equation (6) assumesthat a single laser is being used on each spacecraft; it ispedagogical to consider this simplified case first, but weshall generalize our discussion to the realistic case of sixLISA lasers in Sec. II F.]

Canceling laser phase noise is where interferometrycomes to the rescue. Look at Fig. 2 for combinations ofyij measurements in which two laser pulses arrive simul-taneously at spacecraft 1 at time t, depart simultaneouslyfrom spacecraft 1 at time t, or arrive and depart simulta-neously to and from spacecraft 1 at time t. We subtract theyij measurements, represented graphically by arrows,when they share the same event of emission or reception(i.e., when their arrowtails or arrowheads meet), and weadd them when the receiving spacecraft of one measure-ment is the emitting spacecraft of the other (i.e., whenarrowtail follows arrowhead). In all of these combinations,the laser-frequency noise C1�t� generated at time t onspacecraft 1 is canceled out by entering twice with oppo-site signs; however, GWs are not canceled (not even at timet), because they come into Eq. (5) with differentnmij-dependent projection factors. The combinations ofFig. 2 do still contain frequency noise from lasers 2 and3, and from times other than t; it is however a simple leapto cancel even those by arranging together more measure-

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ments. We then formulate a Geometric TDI principle: toobtain a laser-noise-canceling GW observable, line uparrows (i.e., yij measurements) head to head, tail to tail,or head to tail, creating a closed loop that cancels lasernoise at all pulse emission and reception events. If noarrowhead or arrowtail is left unpaired, the closed looprepresents a linear combination of delayed yij measure-ments that completely cancel the three laser noises Ci�t�.

Remarkably, it is usually possible to interpret eachclosed-loop combination as the interferometric measure-ment performed by comparing the phases of laser beamsthat follow the paths marked by the arrows. Let us see anexample. The arrows of Fig. 3 (left panel) reproduce thepaths followed by light in an equal-arm Michelson inter-ferometer; operating in analogy with Fig. 2, and attributingthe time t to the final common event of reception at space-craft 1, we write the corresponding algebraic expression

y1302�t� L3� � y231�t� � y3201�t� � y123�t� L20 �: (7)

Here the sum of two time-consecutive yij observables, suchas y1302�t� L3� and y231�t� (here L3 is the light-travel timebetween spacecraft 2 and 1), simulates the reflection of thelaser off a mirror: in terms of laser phases, we see that theintegral of this sum reproduces the total phase shift accu-mulated along the path 1! 2! 1. By contrast, the head-to-head difference of two such double arrows simulates aphotodetector: it reproduces the difference of the phaseshifts accumulated along the two paths. All in all, Eq. (7)shows that the combination of four (one-way) yij measure-ments can synthesize the phase-difference output of aMichelson interferometer, as emphasized by Tinto andArmstrong [4], and shown graphically by Shaddock [10]and Summers [14]. Inserting the laser noises Ci�t� inEq. (7), we get

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FIG. 4 (color online). Mapping first-generation TDI observables into closed arrow loops. The Sagnac (�), unequal-arm Michelson(X), and Relay (U) observables of first-generation TDI have a straightforward interpretation as synthesized two-beam interferometers.More interestingly, the beacon (P) and monitor (E) observables can be seen as four-beam interferometers, with the four beamscombining into different pairs at the events of initial emission and final reception. To interpret the symmetrized Sagnac (�) observablesas six-beam interferometers, three different beam pairings must be invoked to explain the cancellation of laser noise at emission andreception and the relative insensitivity of � to GWs (see main text).


�C1�t�L3�L30 � �C2�t�L3�� C2�t�L3��C1�t��

� �C3�t�L20 � �C1�t�� C1�t�L20 �L2�

�C3�t�L20 ��; (8)

which sums up to zero in interferometer geometries where4

L3 � L30 � L20 � L2: our equal-arm Michelson combina-tion is then truly laser-noise canceling. It is, however,sensitive to GWs, as can be seen by inserting Eq. (5) inEq. (7).

More generally, we can set three simple rules to turn aclosed arrow loop into a combination of yij measurementsthat cancels laser noise:

(1) s

4If tintroduLl0 in t

tart at any spacecraft, and write down the appro-priate yij for each arrow, following the loop (goingalong or against the direction of each arrow) until allarrows are used up (if more than two heads or tailsmeet at any spacecraft, different visiting orders willyield different observables);

(2) u
se a plus (minus) sign for arrows followed along(against) their direction;
(3) g
ive time arguments to the yij, remembering thatmeasurements are always made at the receivingspacecraft (at the arrowhead); use the nominaltime t for the first yij, and then add (subtract) theappropriate Ll for each arrow followed along(against) its direction.
C. The observables of first-generation TDI

The first laser-noise-canceling combinations for LISAwere discovered using an algebraic (rather than geometric)

he interferometer is rotating, the Sagnac effect [21]ces a distinction between the light-travel times Ll andhe two directions [8,9].

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approach, matching up delayed yij measurements in such away that all laser-noise terms would cancel. Using thisprocedure, Tinto, Armstrong, and Estabrook [4–6] ob-tained expressions for first-generation TDI observables,which cancel laser noise in static unequal-arm geometries.These observables are sums of either six or eight delayedyij measurements (for short, links). See Fig. 4.

The 6-link observables �, �, � (mapped into each otherby relabeling the spacecraft cyclically) use all six LISAoriented arms and measure the phase difference accumu-lated by two laser beams traveling around the LISA arrayin clockwise and counterclockwise directions: thus, theybehave much like a Sagnac interferometer and are knownas Sagnac observables. A related 6-link combination, thesymmetrized Sagnac observable � , has the useful propertyof being relatively insensitive to GWs in the low-frequencylimit.5

The 8-link observables X, Y, Z (also mapped into eachother by cyclic spacecraft relabelings) use two of the LISAarms in the two directions. They are unequal-arm general-izations of the Michelson observable of Eq. (7): for un-equal arms, the latter would fail to cancel the laser-noiseterms from the tails of the two paths, because L3 � L30 �

L20 � L2. The solution is to have both paths go througheach arm once (hence the eight terms), building up thesame light-travel time (see the right panel of Fig. 3).Related 8-link combinations, known as observables ofthe U, P, and E type, use different sets of four orientedarms out of six and have GW sensitivity comparable to theMichelson combinations [5,6].

Prior to my work, it was unclear whether the P-type andE-type observables could be interpreted as synthesized

5Most interestingly, a GW-insensitive observable allows theobservational distinction of a stochastic GW background frominstrumental noise [22].

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interferometric observables.6 In Fig. 4, we show that this ispossible if we identify four distinct laser beams, paired inalternative ways to cancel laser noise at the path tails (dots)and path heads (ending arrows). The two path origins arenot simultaneous and neither are the two path endings.

The symmetrized Sagnac observable � , which also de-fies explanation as a two-beam synthesized interferometer,can be interpreted as a six-beam interferometer, wherebytwo different pairings explain the cancellation of lasernoise at emission (dots) and reception (arrows). Yet an-other pairing, shown by the thin diagonal lines in Fig. 4,explains why � is relatively insensitive to GWs at lowfrequencies: in the limit of equal arms, each pair of parallelarrows represents the difference of two symmetric mea-surements yij�t� and yji�t� that share the same times ofpulse emission and reception. Taylor-expanding thehTTij �. . . ; t� terms of Eq. (5) around t and around either pl1or pl2, we find that yij�t� � yji�t� / L3h000ij . By contrast, thedifferences of head-to-tail double arrows that appear in Xsum up to L2h00ij. Considering monochromatic GWs offrequency fGW, we see that the GW response is smallerfor � than for X by a factor7 2�fGWL ( ’ 0:1 for fGW �10�3 Hz, ’ 0:01 for 10�4 Hz). Since the response to theLISA secondary noises is approximately the same for �and X [as can be seen using Eqs. (20), discussed below], �turns out to be relatively insensitive to GW.

D. The observables of second-generation TDI

This interpretation of TDI observables as 2N-beam syn-thesized interferometers is intriguing, but also troubling,since it casts a suspicion of arbitrariness on the selection ofa standard set of observables, and it complicates exploringthe space of all possible combinations. Fortunately, theapplication of the tools of modern algebra to TDI showedthat all first-generation observables can be obtained asalgebraic combinations of four generators [7]. This ap-proach was extended [7] to modified TDI observables [8–11], which cancel laser noise in rotating LISA geometries,where the Sagnac effect [21] introduces a distinction be-tween light-travel times in the two directions. (TheMichelson-, U-, P-, and E-type observables of first-generation TDI are bona fide modified TDI observables,if written with the correct primed and unprimed delayindices; by contrast, the Sagnac observables of modifiedTDI are different and twice as long as those of first-generation TDI.)

However, the algebraic approach cannot be extendedeasily to the observables of second-generation TDI, whichcancel laser noise in LISA geometries with time-dependent

6The interpretation of U as a synthesized observable wasalready clear to F. B. Estabrook (unpublished note).

7For unequal interferometer arms, � / L��L�h00ij, so the ratiobetween the � and X responses is ��L=L� 0:01.

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arm lengths.8 As pointed out by Cornish and Hellings [9],in this situation it is necessary to keep track of the order ofretardations: for instance, the unequal-arm Michelsoncombination of Fig. 4 would translate to

y1302;3220 �t� � y231;220 �t� � y123;20 �t� � y3201�t�

� y231�t� � y1302;3�t� � y3201;303�t� � y123;20303�t�;

(9)

where, using the semicolon notation of Ref. [10],

yilj;r1�t� � yilj�t� Lr1�;

yilj;r2r1�t� � yilj�t� Lr1 � Lr2�t� Lr1��; (10)

and so on: the nominal time t is delayed incrementallystarting from the rightmost delay index r1. (A similarnotation with commas instead of semicolons is usedwhen the arm lengths are constant and the order of theretardations is not important.)

Inserting the laser noises Ci in Eq. (9), we see that theycancel in pairs, except for the terms from the tails of thetwo paths,

C1;303220 � C1;220303�t� ’ _C1�t��t;303220 � t;220303�; (11)

Taylor-expanding the retardations to first order and keep-ing only linear terms in the _Ll�t�, we get

_C 1�t��L20 �L2�� _L3� _L30 ��L3�L30 �� _L20 � _L2��; (12)

where all the Ll and _Ll are implicitly evaluated at time t.(More generally, each retardation index ri generates aresidual term proportional to Lri

_Lrj for each index rj toits left.)

In short, much like what happened with the simpleMichelson combination [Eq. (7)] for unequal-arm geo-metries, a laser-noise residual appears in Eq. (12) becausethe light-travel times built up along the two interferingpaths are different; graphically, the tails of the twopaths do not match precisely. This is because, althoughboth paths contain the same set of links, they do so indifferent orders, and the retardations do not commutewhen the arm lengths are time dependent. (This is alsothe reason why the algebraic approach becomes arduousfor second-generation TDI, where it involves the solutionof polynomial equations for noncommuting variables.)As in the upgrade from equal-arm to unequal-arm(first-generation) Michelson observables, one solution isto compose the two paths so that each goes through each

8With second-generation TDI, the cancellation occurs up to(and including) terms proportional to Ll _Lm, which is more thansufficient for realistic LISA spacecraft orbits.

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arm twice, in different orders [10]. The residual of theresulting 16-link combination vanishes up to the firstTaylor order and to the first degree in _Ll (henceforth, to

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first order/degree). This second-generation unequal-armMichelson observables (known as X1) may be written inour notation

y1302;3220220303�t��y231;220220303�t��y123;20220303�t��y3201;220303�t��y123;20303�t��y3201;303�t��y1302;3�t��y231�t��y3201�t�

�y123;20 �t��y231;220 �t��y1302;3220 �t��y231;303220 �t��y1302;3303220 �t��y3201;303303220 �t��y123;20303303220 �t�; (13)

which is related to the X1 defined in Refs. [10,11] by achange of sign and by the use of the opposite conventionfor primed and unprimed indices.

Second-generation generalizations of all first-generationTDI observables were described by Tinto and colleagues[10,11]. For the analogs of the �, �, �, and � observables(which are formally identical to their modified TDI coun-terparts [8,9], except for the interpretation of the delayindices as noncommuting), laser-noise cancellation is notcomplete, even to first order/degree: however, the residualsconsist of symmetric sums of Ll _Lm terms that turn out to besmall for realistic LISA orbits.

E. The combinatorial enumeration of TDI observables

Our geometric approach to TDI makes it possible toenumerate all the second-generation TDI observables ofgiven length. The key to this is the Feynman-Wheeler [23]Geometric TDI principle: any 2N-beam Geometric TDIclosed loop can be seen as a single beam that travelsforward and backward in time to meet itself back at itsorigin.

For instance, the two-beam equal-arm Michelson com-bination of Fig. 3 (left panel), can be interpreted as a singlebeam that departs at the initial time t� L3 � L30 , travelsforward in time to be measured at time t� L3, and againtravels forward in time to be measured (and interfereagainst itself ) at time t; the beam then moves backwardin time to be emitted at time t� L20 , and again movesbackward in time to be emitted at the original time t�L20 � L2 (equal to t� L3 � L30 , since the arm lengths areequal). This closes the loop and cancels laser noise at alljunctions (when we translate graphs to formulas, we mustremember to give minus signs to all the backward-timearrows, drawn dashed in Fig. 3).

Once we have established that all n-link loops can berepresented as a single loop, we can enumerate themcombinatorially by choosing a starting spacecraft and, forn times over, choosing the future or past time direction, andthe leftward (clockwise) or rightward (counterclockwise)movement direction, in all possible combinations. Eachloop can be denoted by the index of the initial spacecraft,followed by a string of ‘‘L’’ or ‘‘R’’ crested by ‘‘!’’ forforward-time arrows and by ‘‘ ’’ for backward-time ar-rows; this notation is translated easily into strings of linkindices crested by their time directions (henceforth, link

strings). For instance, we would write �1�LR��!

LR �� 303

��!202 ��

and �1�LRRL��!

RLLR ��

303220��!

330202 ��

for the loops in theleft and right panels of Fig. 3, respectively.

Not all 3� 22n strings with n links correspond to laser-noise-canceling combinations, because the total light-travel time accumulated across the loop must be zero (forsecond-generation TDI, zero to first order/degree).However, it is quite straightforward to set simple closurecriteria that identify the true TDI combinations:

Pre-TDI interferometry.—For equal-arm geometries, theloop must end at the initial spacecraft (#�L

!; R � �

#�R!; L � mod 3 � 0, where # denotes the number of oc-

currences of a symbol in the string), and it must have a nulltotal light-travel time (#�L

!; R!� � #�L

; R �). We denote

the combinations that satisfy this property as closed.First-generation TDI.—For unequal-arm geometries

with generic Ll � Ll0 , the loop must end at the initial

spacecraft and satisfy #� l!; l0!� � #� l

; l0 � (for l �

1; 2; 3), which yields a null total light-travel time. Wedenote the combinations that satisfy this property as jLjclosed.

Modified TDI.—For unequal-arm geometries with ge-neric Ll � Ll0 , the loop must end at the initial spacecraftand satisfy #� l

!� � #� l

� (for l � 1; 10; 2; 20; 3; 30), which

yields a null total light-travel time. We denote the combi-nations that satisfy this property as L closed.

Second-generation TDI.—For unequal-arm geometrieswith generic, time-dependent Ll�t� � Ll0 �t�, first-order/degree laser-noise cancellation is obtained for loops thatare L closed, and in addition satisfy #� l

!_m!; l

_m � �#� l!

_m ; l

_m!� (with l;m � 1; 10; 2; 20; 3; 30), where a pairl _m is counted for each link l

!with itself and with all the

links m! and m to its right, and for each link l

with all thelinks m! and m to its right. This condition yields a totallight-travel time that is null to first order/degree (see

Appendix A 1). For instance, for 303��!

202 ��

we count 3030��!

,

303��!

, 30!

20

, 30!

2

, 33�!

, 3!

20

, 3!

2

, and 2020 ��

; hence thecounting does not satisfy the property given above. Wedenote the combinations that satisfy this property as _Lclosed.

The closure criteria induce useful symmetry propertiesfor the link strings:

Null bigrams.—The bigrams L!

L

, R!

R

, L

L!

, andR

R!

(or equivalently l!

l

and l

l!

) always representcombinations of two yij measurements that sum up to

-7


exactly zero: any n-link string that contains such bigramsrepresents a combination of length smaller than n.

Cyclic string shift.—Shifting a string cyclically pro-duces a combination that has the same closure propertiesas the original, and that differs only by an overall timeadvancement or retardation, and by an advancement orretardation applied selectively to the shifted terms, butconsidered negligible at that closure level (for instance,for an L-closed loop the additional selective retardationwould be of leading order/degree Ll _Lm; for an _L-closedloop, of a higher order/degree).

Time and direction reversal.—Simultaneously swappingthe primedness and time direction of all link indices pro-duces a combination that has the same closure properties asthe original, and that differs from the original by itshandedness.

Cyclic index shift.—Shifting link indices cyclically(1! 2; 2! 3; 3! 1) produces a combination that hasthe same closure properties as the original, and that differsonly by a relabeling of indices. Noncyclic index permuta-tions, on the other hand, produce illegal strings (i.e., un-connected loops) unless all indices change in primedness,in time direction, or in a combination of the two.9

String reversal.—Reversing a string while swapping all‘‘!’’ and ‘‘ ’’ symbols (i.e., reversing all time directions)produces a combination that has the same closure proper-ties as the original, and that differs only in sign and by anoverall time advancement or retardation considered negli-gible at that closure level.

Splicing.—Inserting any link string at any compatiblepoint within a link string yields another legal link string.Here compatibility means that the spacecraft visited by thefirst loop at the insertion point (for instance, after 1

!,

spacecraft 2; after 1

, spacecraft 3) must be the same asthe initial spacecraft of the inserted loop. The resultingstring has at least the closure properties shared by thespliced fragments.

For instance, the unequal-arm Michelson (L-closed)string

130231232

0��!1323

0120321

��(14)

(where the subscripts show the spacecraft visited by theloop before and after traversing each link) can be spliced atits center with its own L-closed reversal,

130231232

0��!�1232

0130231

��!20321323

01�

��323

0120321

��; (15)

yielding an _L-closed loop that is in fact the second-generation TDI Michelson observable.

9The reflectionlike symmetry considered below Eq. (36) con-sists in exchanging two indices and swapping their primedness,while reversing the time direction of the third index.

042003

At this point it is also useful to give a rule to translatelink strings into yij combinations:

(1) S

-8

tarting at the left end of the string, write a yijmeasurement for each index, according to the re-placement rules f1; 2; 3; 10; 20; 30g fy32; y13; y21;y23; y31; y12g; attribute a plus sign for ! links anda minus sign for links;

(2) w
hile doing this, build the delay sequence to beapplied to each new yij, adding (from the left) anadvancement index r before translating each r!, anda retardation index s after having translated each s .
For instance, the 303220��!

330202 ��

string would translate to

y1302;30

�t� � y231;330

�t� � y123;2330

�t� � y3201;202330

�t�

� y231;202330

�t� � y1302;3202330

�t� � y3201;303202330

�t�

� y123;20303202330

�t�: (16)

To accommodate advancement indices, the semicolon no-tation of Eq. (10) is extended by the advancement rule

yilj;a�t� � yilj�t� �a�; (17)

with �a defined as the time experienced by light propagat-ing along link a for emission at time t, and given implicitlyin terms of La�t� by �a�t� � La�t� �a�t��. Retardationsand advancements are applied incrementally starting fromthe rightmost index.

Equation (16) appears more complicated than Eq. (9),but it encodes essentially the same yij combination: the twoequations are related by a time retardation, as can be seenby evaluating Eq. (16) at the time t;303220 . This adds 303220

to the right of the delay sequence for each yij, and sinceadjacent pairs ll and ll cancel in delay sequences (by thevery definition of Ll and �l), Eq. (16) then turns intoEq. (9). Conversely, Eq. (13) may be obtained by applying

the translation rule to the string 303220220303��!

202330330202 ��

and evaluating the resulting expression at the timet;330202202330 . A slightly more complicated version (seeApp. A 2) of the rule given above yields yij combinationsthat are closer to standard TDI notation.

F. Extension of Geometric TDI to six-laser LISA con-figurations

The extension of our discussion to LISA configurationswith six lasers introduces three additional laser phase noisevariables C�1�t�, C

�2�t�, and C�3�t�, corresponding in Figs. 2

and 3 to the optical benches on the right-hand side of thespacecraft (if we look toward the center). Equation (6)changes accordingly:

TABLE I. Geometric survey of second-generation TDI observables. Quasiduplicates that differ only by a sign or by a timeadvancement are not counted. The X, U, E, and P types represent generalizations of the 8-link observables of the same name:X-type (Michelson) observables use two arms in both directions, U-type observables use four oriented arms in a relay configuration,E-type and P-type use four oriented arms in beacon and monitor configurations; the observables reported under other use either five orsix oriented arms. The number of beams corresponds to the number of contiguous substrings with the same time direction, minimizedwith respect to cyclic string shifts. All the observables tallied in this table are available in extenso at the web page www.vallis.org/tdi.

Links Comb. space Unique obs. X type P;E type U type other 2-beam 4-beam 6-beam 8-beam 10-beam and higher

16 4� 109 48 12 18 18 0 3 27 0 18 018 7� 1010 192 0 12 12 168 6 24 102 60 020 1� 1012 660 24 18 18 600 12 114 276 90 16822 2� 1013 2412 0 36 36 2340 30 264 732 792 59424 3� 1014 12585 144 90 90 12261 99 945 2676 4566 4299


yij � C�i;l � Cj � GWs for unprimed l�i; j�;

yij � Ci;l � C�j � GWs for primed l�i; j�;

(18)

here unprimed and primed link indices l�i; j� correspond tocounterclockwise and clockwise yij measurement direc-tions, respectively. As shown in Ref. [11], all argumentsand derivations valid with three lasers can be applied to asix-laser configuration by replacing all yij with y�6�ij (�ij inRef. [11]) defined by

y�6�ij � yij �1

2�zlj;l � zij;l� for unprimed l�i; j�;

y�6�ij � yij �1

2�zjljj � zij� for primed l�i; j�;

(19)

where the zij�t� are the back-plane measurements compar-ing the phases of the two lasers on each spacecraft, andwhere the notation jlj removes a prime from l, if present.Because zij�t� � ��C

�j �t� � Cj�t�� [for unprimed and

primed l�i; j�, respectively], the C�j disappear from

Eq. (18), and Eq. (6) is restored for the y�6�ij . This justifiesall the developments reported in this paper also for six-laser LISA configurations.

It should be mentioned in this context that the phasenoise from the random motion of the optical benches entersthe y�6�ij with the same time signature as the laser phasenoises and is therefore also canceled by TDI. The LISAsensitivity to GWs is then set by the remaining secondarynoises. Adopting the schematization of the measurementprocess used in most of the TDI literature, and the notationused to describe the Synthetic LISA simulator [20], theresponse of the y�6�ij to the secondary noises is given by

y�6�ij � yopij � 2pmj�pm�i;l�pmi;l for unprimed l�i; j�;

y�6�ij � yopij �pm

�j �pmj for primed l�i; j�; (20)

where yopij is the optical-path noise in the yij phase mea-

042003

surement, and pmi and pm�i are the velocity noises of thetwo proof masses aboard spacecraft i. Because pmi;l �pmi and pm�i;l � pm

�i have the same time signature as laser

phase noises, they are canceled in TDI observables; thus,all retardations can be removed from the unprimed-l�i; j�expression of y�6�ij , casting it to the same form as itsprimed-l�i; j� counterpart.

III. A GEOMETRIC SURVEY OF SECOND-GENERATION TDI OBSERVABLES

I have written a computer program to list all the second-generation TDI observables consisting of 24 or fewer yijmeasurements. For each even length n, this was achievedby enumerating all 22n possible LR strings, and checkingeach of them for _L closure, according to the counting rulegiven in Sec. II E. Already for 24-link strings, the combi-natorial space is huge, and an exhaustive search requiredmore than 10 000 CPU hours. The resulting list of observ-ables was then reduced to a minimal set by removing all thequasiduplicates that differ only by a sign or by a cyclicstring shift. I have kept as distinct the observables thatdiffer by a cyclic index shift (in first-generation TDI, thiswould correspond to counting X, Y, and Z as separateobservables). The reduced list of TDI observables is avail-able at the web page www.vallis.org/tdi, annotated withtheir temporal footprint (see Sec. III B), number of beams,type, and splicing composition (see Secs. III A and III C ).

My results are tallied in Table I. Here the X, U, E, and Ptypes represent generalizations of the 8-link observables ofthe same name: X-type (Michelson) observables use twoarms in both directions, U-type observables use four ori-ented arms in a relay configuration, E-type and P-type usefour oriented arms in beacon and monitor configurations;the observables tallied under other use either five or sixoriented arms. Here are the highlights of the survey, whichare discussed in more detail in the following sections.

(i) I

-9

find that the shortest second-generation TDI ob-servable has length 16. By contrast, modified TDIobservables begin at length 8.


(ii) I

11Neglables w

10In faobservacellationdistinctnoise ismeasuretemporamind thRef. [11

recover all10 the known 16-link second-generationTDI observables, previously obtained by Tinto andcolleagues [10,11] by applying commutatorlike de-lay operators to the 8-link observables of modifiedTDI. From a geometric viewpoint, all the 16-linkobservables can be understood as self-splicings ofthe 8-link observables of the same type. This showsthat the former reduce to finite time differences ofthe latter, up to time shifts of the first order/degree.It follows that the second-generation TDI observ-ables have the same sensitivity of the modified TDIobservables of the same type,11 not only in theequal-arm limit, but unconditionally.See Sec. III A for details.

(iii) I
n addition, I obtain alternative forms of the known16-link observables. The alternative forms use alarger number of beams (e.g., four beams for X,as opposed to the standard two), or a differentallocation of links in the beams (e.g., 5� 4� 3�4 or 6� 4� 2� 4 for U, as opposed to the stan-dard 7� 4� 1� 4 structure). The alternativeforms, too, can be understood as self-splicings ofthe 8-link modified TDI observables of the samekind.The alternative forms have the same sensitivity toGW signals as the original forms in idealized mea-surement conditions, but they can improve on themwhen realistic aspects (such as quantization of thephasemeter output and technical noises) are takeninto account. In addition, the alternative forms havea reduced temporal footprint (the difference be-tween the times of the earliest and latest phasemeasurements involved in their construction); thisfeature can be advantageous in the presence of gapsor glitches in the yij data, because it reduces theextent of defect propagation to the TDI time series.See Secs. III A and III B for details. Appendix A 4gives explicit algebraic expressions for all the 16-link observables in terms of the yij measurements.
(iv) S
econd-generation TDI observables are found inincreasing numbers at lengths 18, 20, 22, and 24. Aminority are of the X, U, E, or P types, while mostuse either five or six oriented arms.All 18- to 24-long observables can be understood as
ecting of course the fact that the modified TDI observ-ould not cancel laser noise in a flexing LISA.

ct, our U-type combinations do not include the U1ble given in Ref. [11], which achieves laser-noise can-

at the approximate time t� 4L through the sum of fouryij measurements; by contrast, in Geometric TDI laseralways canceled by construction between pairs of phasements. However, the U1 of Ref. [11] has almost the samel structure as our 1012030

��!1

2030101��!

3020101103020 ��

(keep inat the primedness of our indices is the opposite of]).

042003-10

splicings of modified TDI observables of length 8to 18, sometimes with the inclusion of null bi-grams; most, but not all, are self-splicings. I con-jecture that all second-generation TDI observablesof any length can be generated as splicings of twomodified TDI observables.See Sec. III C for details.

(v) U
p to length 24, I do not find any _L-closed observ-ables of the � type (defined as having suppressed,but nonzero, GW response at low frequencies). Iconjecture that the � type is incompatible with _Lclosure. This does not exclude the existence ofnon- _L-closed �-type observables (such as the �1,�2, and �3 described by Tinto and colleagues [11])that do not cancel laser noise to first order/degree,but bring it sufficiently below the LISA secondarynoise to be useful in practice.
A. Structure and sensitivity of the 16-link observables

The standard second-generation TDI X observable is

X16;21 : 303220220303

��!202330330202 ��

�2 beams�; (21)

which is related to the X1 defined in Eq. (13) by X16;21

X1;330202202330

. The alternative forms found in our geometricsurvey are

X16;4;�11 : 303220

��!330202330 ��

220303��!

202 ��;

X16;4;�11 : 303

��!202330 ��

220303220��!

330202 ��

�4 beams�;(22)

which differ between themselves only by handedness, and

X16;4;01 : 220220

��!330202 ��

303303��!

202330 ��

�4 beams�; (23)

which turns out to have vanishing response in the equal-arm limit to both noise and GWs, at all frequencies. To seethat X1 and the X16;4;�1

1 have all the same GW sensitivity asthe 8-link modified TDI X (neglecting of course the factthat X would not cancel laser noise in a flexing LISA), wereason as follows.

As we have learned in Eq. (15), X1 can be interpreted asa self-splicing of X with its reversal. If we take X to bedefined by Eq. (9), we see that

303220��!

330202 ��

X;202330

; 220303��!

202330 ��

�X;330202

:

(24)

Since the time at the splicing point in 303220��!

330202 ��

ist;202330

, from Eq. (15) we see that

303220220303��!

202330330202 ��

’ X;202330

� X;330202202330

; (25)

here the symbol ‘‘’’’ denotes equality up to selectivedelays or advancements of order Ll _Lm in the yij, notspecified by the formal delay strings. (In this case, these

spurious delays appear because 220303��!

202330 ��

is only L


closed, so the time at the beginning and at the end of theinserted string is different by terms of order Ll _Lm; con-

sequently, the last four yij observables of 303220��!

330202 ��

arereally evaluated at the time t

;303220330202202330, not just t

;202330.)

Rewriting Eq. (25) in terms of X1 and reabsorbing thetime advancements by evaluating the equation at timet;303220220303, we find

X1 ’ X;220303 � X: (26)

Thus, up to delays of first order/degree, self-splicingsproduce finite differences of observables. (Indeed, it is awell-known fact in the literature on second-generation TDIobservables that the standard 16-link observables are ap-proximately equal to finite differences of the standard 8-link observables of the same type.)

Now, because the individual yij measurements respondlinearly12 to GWs and to all instrumental noise sources, thestrain sensitivity of X to monochromatic sources of fre-quency f at a given sky position is proportional to~Xn�f�= ~XGWs�f�, where ~Xn�f� is the (square-root) spectraldensity of noise in X, and ~XGWs is the Fourier transform ofthe GW response function. The constant of proportionalityis SNR=

��Tobsp

, where SNR is the fiducial signal-to-noiseratio at which the sensitivity is defined, and

��Tobsp

is thetime duration of the observation. Combining the Fourier-transform time-shifting property with Eq. (26), and con-sidering that first order/degree terms can be neglected forsecondary noises and GWs (which are much weaker thanthe laser phase noises), we see that X1 must have the samesensitivity as X:

~Xn1�f�~XGWs1 �f�

��e2�if�t � 1� ~Xn�f�

�e2�if�t � 1� ~XGWs�f�’

~Xn�f�~XGWs�f�

; (27)

with �t � t;220303 � t. This is true generically for anyspacecraft geometry, and not just in the equal-arm limit.

To see that X16;4;�11 and X16;4;�1

1 , too, have the samesensitivity to GWs as X (in the limit of perfect laser phasenoise cancellation), it is then sufficient to show that theyare self-splicings of X and of its (cyclically shifted) rever-sal:

X16;4;�11 : 303220�330

��!202330 ��

j220��!�303202 ��

;

X16;4;�11 :303

��!�202330 ��

j220303�220��!

330202 ��

:(28)

Moving on to the U type, we see (for instance) that thethree second-generation U-type observables that use theoriented arms 1, 10, 2, and 3,

12Linearity is always assumed in the model of measurementused to derive TDI. Significant nonlinearity in the phase mea-surements has been explored little, but would probably provevery detrimental to the delicate cancellation of laser phase noiseachieved by TDI.

042003

32110��!

231�23 ��

j10132��!

101�10 ��

;

32110��!

23 ��10132��!

10123�110 ��

; 32110�132��!

j10123 ��

10!�23110 ��

;

(29)

are generated by the self-splicings of the modified TDIU-type observable that uses the same oriented arms,

32110��!

23110 ��

. Thus, the modified TDI and second-generation TDI U-type observables have the same GWsensitivity (in the limit of perfect laser phase noise cancel-lation). Similar arguments hold for the P- and E-typeobservables.

Altogether, we find empirically that all the 16-linksecond-generation TDI observables can be generated asself-splicings of the 8-link modified TDI observables of thesame kind. Conversely, it can be proved that all self-splicings of L-closed observables are _L closed (seeAppendix A 3).

B. Advantages of the alternative 16-link observables

As mentioned above, the alternative forms of the 16-linkobservables can have a smaller time footprint than thestandard forms. For instance, the standard X1�t� ofEq. (13) involves 16 yij measurements taken13 within theinterval �min�t;303220220303; t;220303303220 �; t�, for a time span ’8L, and each single yij measurement appears in X1 at timesdisplaced by as much as ’ 6L. Thus, X1 will be unavailableduring the first and the last ’ 136 s (i.e., 8L) within eachLISA data-taking period. Moreover, a data gap in a singleyij measurement will appear in the X1 time series at fourdistinct times spanning ’ 6L. By contrast, the alternativeforms X16;4;�1

1 involve 16 yij measurements taken within atime interval of span ’ 6L, and the single yij appears attimes displaced by at most ’ 4L. The gain is significant, ifnot dramatic.

The alternative forms for theU-, E- and P-type variablesalso yield footprint gains with respect to their standardforms (from ’ 7L to ’ 5L for U and from ’ 5L to ’ 4Lfor E and P). These gains are possible because the alter-native observables are obtained, loosely speaking, by fold-ing the standard versions in time, using both timeadvancements and retardations, as opposed to retardationsonly, to arrange the yij measurements so that laser phasenoise is canceled at all emission and reception events.

Another advantage of the alternative forms is that theycan yield an improvement in GW sensitivity in realisticmeasurement conditions. In the top panel of Fig. 5, thedashed curve shows the power spectral density (PSD) ofsecondary noise for the standard X1 observable, drawn in

13Although the nominal times of the yij are displaced only by’ 7L, we must remember that in a six-laser LISA configurationsome of the y�6�ij contain one additional delay, as given byEq. (19).

-11

FIG. 5. TDI response of the second-generation X-type varia-bles to the fundamental secondary noises, according to Eq. (20),assuming equal arms and proof-mass optical-path noise spectraldensities given by Spmi � 2:5� 10�48�f=Hz��2 Hz�1 and Sopi �1:8� 10�37�f=Hz�2 Hz�1 (following Ref. [20]). The top panelshows the noise response of the standard 16-link X1 observable,as compared to the response of the alternative 16-link forms ofEq. (22), which have half as many nulls. The middle panel showsthe noise response of the two-beam, 20-link X1 observables ofEq. (30). The bottom panel shows the three new noise responsesfound for 24-link X1 observables (solid and dashed: self- andnon-self-splicings of 12-link observables; dashed-dotted: self-splicings of 8-link observables, with inclusions). To avoid visualclutter, two of the 24-link X1 curves are not plotted beyond f �1=�2L� (shaded region).

14Roughly speaking, the GW transfer function for the standardX1 is found by inserting Eq. (5) into Eq. (13) and Fourier-transforming, assuming a monochromatic source at a fixed skylocation. The resulting transfer function is usually integratedover sky locations.

15However, the simulations of Ref. [24] seem to suggest thatuncanceled laser noise, under the assumption of perfectly linearresponses, will display the same nulls as the fundamental sec-ondary noises.

16In the equal-arm limit, Sorig;16n � 4 cos2�2�fL�Salt;16n .17As discussed in footnote 10, the observables are not identical:

however, they do have the same secondary-noise PSD and GWtransfer function.


the limit of equal arm lengths. Following Ref. [20], weassume that secondary noise consists entirely of proof-mass noise (idealized as stationary and Gaussian, withPSD Spmi � 2:5� 10�48�f=Hz��2 Hz�1) and of optical-path noise (also stationary and Gaussian, with PSD Sopi �1:8� 10�37�f=Hz�2 Hz�1). As discussed around Eq. (27),the sensitivity to GWs (averaged over noise realizations) iscomputed by dividing the secondary-noise rms power by

042003

the GW transfer function.14 See, for instance, Ref. [6] forplots of the GW sensitivities of the first-generation X-, U-,P-, and E-type observables, common also (as discussedabove) to the second-generation observables of the sametype.

Although the noise PSD has nulls at multiples of theinverse arm length light-travel time [for the standard X1,the nulls are at f � k=�4L�], the sensitivity to GWs re-mains finite in idealized conditions, because the GW trans-fer function displays zeros of the same order at the samefrequencies. In reality, we should expect a degradation ofsensitivity at these frequencies, because noise as a wholecan only drop to the level of uncanceled laser noise,15 or ofother technical noises (such as quantization noise), effec-tively filling in the nulls. The nulls are unwelcome alsobecause they imply that a very large dynamic range isneeded for sensitive measurement at those frequencies.The alternative forms X16;4;�1

1 improve on this situation,because their noise PSD and GW transfer functions16 havehalf as many nulls [at f � k=�2L�], as shown by the solidcurve in the top panel of Fig. 5.

Similar gains are found for the other 16-link second-generation TDI observables. The standard U-type observ-able of Ref. [11], which we may represent as17

32110132��!

10123 ��

10!

23110 ��

, has nulls at f � k=�3L�, while

the alternative form 32110��!

23123 ��

10132��!

10110 ��

has nulls onlyat f � k=L. The distribution of the nulls improves also forall the alternative forms of the E- and P-type observables.

C. Longer observables

Although the size of the combinatorial space of LRstrings scales as 22n with increasing string length n, wefind empirically that the number of valid _L-closed combi-nations grows only as 2n (at least up to n � 24); by con-trast, the number of the less constrained L-closedcombinations grows roughly as 23n=2 (at least up to n �16).

The majority of the longer second-generation observ-ables use either five or six LISA oriented arms, and there-fore do not belong to any of the X, U, E, or P types. We dohowever find new forms for these. Throughout this section,

-12


we shall consider examples of the X type: findings andconclusions are similar for the other types.

At length 20, we find two X-type observables with twobeams,

303220220�220303��!

202330�202330202 ��

;

303220303�220303��!

202330�330330202 ��

;(30)

which can be interpreted as the self-splicing of the 8-link

modified TDI observable 303220��!

330202 ��

with its reversal

220303��!

202330 ��

, after the insertion of the double null bigrams

220��!

202 ��

and 303��!

330 ��

[shown in bold in Eq. (30)]. Because theyij measurements corresponding to the null bigrams sumup to zero to first order/degree, we can repeat the argu-ments of Sec. III A to see that the 20-link X1 observables ofEq. (30) have the same sensitivity as the standard 16-linkform. Moreover, reasoning along the lines of Appendix A3, we can prove that the self-splicings of L-closed stringsare _L closed even with the inclusion of null bigrams.

At length 20 we find also alternative X1 observables withfour beams,

303220220��!

�330 ��j220303��!

202�202330202 ��

;

303220303�303��!

202330 ��

j220��!�330330202 ��

;

303303�220303��!

202330�330 ��

220��!

330202 ��

;

303220��!

330 ��

220�220303��!

202330�202202 ��

;

(31)

and with six beams,

303220��!

330 ��

220��!�303 ��j220303��!

202�202202 ��

;

303220��!

330330�330 ��

j220303��!

202 ��330��!

202 ��;

(32)

all of which are again self-splicings of 303220��!

330202 ��

withtwo null-bigram inclusions. The secondary-noise PSD ofthe two-beam, 20-link X1 observables of Eq. (30) is shownin the middle panel of Fig. 5, and has nulls at f � k=�6L�.This PSD is not found for any of the 16-link X1 observ-ables. The four-beam observables have the secondary-noise PSD of the standard 16-link X1 (the dashed curvein the top panel of Fig. 5), while the six-beam observableshave the PSD of the alternative 16-link X1 (the solid curvein the top panel of Fig. 5).

We find more X-type observables at length 24. Some ofthese, such as

303220220220�220303��!

202330�202202330202 ��

; (33)

are self-splicings of the 8-link modified TDI X with qua-druple null-bigram inclusions and have the same GWsensitivity as the standard 16-link X1. They can have thesame secondary-noise PSDs as the 16- and 20-link X1

observables, or a new PSD (the dash-dotted curve in the

042003

bottom panel of Fig. 5) with nulls at f � k=�4L� and atfrequencies given by third-degree algebraic numbers.

Other 24-link X-type observables, such as

303220220�220220303��!

202202330�330202202 ��

;

303303220�220303303��!

202330330�330330202 ��

;(34)

are self-splicings of the 12-link modified TDI X-type ob-

servables 303220220��!

330202202 ��

and 303303220��!

330330202 ��

. The24-link observables of Eq. (34) can be shown to have thesame GW sensitivity as the standard 16-link X1 in theequal-arm length limit. To see this, we parse the 8- and12-link X-type variables as self-splicings of the pre-TDI

(simply closed) observable 303��!

202 ��

, respectively, withoutand with double-null-bigram inclusions:

303�220��!

330�202 ��

;

303220�220��!

330�202202 ��

; 303303�220��!

330�330202 ��

:(35)

If the arm lengths are equal, the yij measurements from thedouble bigrams shown in bold sum up to zero and give nocontribution to the GW and secondary-noise responses; ifthe arm lengths are different, the presence of the bigramsintroduces a preferred direction that distinguishes the sen-sitivities of the 8-link and 12-link observables. In the equalarm length limit, the observables of Eq. (34) can have thesame secondary-noise PSD as the standard 16-link X1

observables, or they can have two new PSDs (shown assolid and dotted curves in the bottom panel of Fig. 5) withnulls at f � k=�2L� and f � �2k� 1�=�4L�, and at f �k=�4L� and f � k=�6L�, respectively.

At length 24 we find also some X-type observablesderived from non-self-splicings of 12-link modified TDIX-type observables, such as

303220220�220303303��!

202330330�330202202 ��

;

303303220�220220303��!

202202330�330330202 ��

:(36)

These non-self-splicings occur between 12-link L-closedlink strings that differ by a reflectionlike, noncyclic indexshift (e.g., 2$ 30, 3$ 20, with 1 and 10 reversing timedirection, if they are present in the string). In the equal-armlength limit, these observables have again the same sensi-tivity as the standard 16-link X1, and they have the samesecondary-noise PSDs as either the standard 16-link X1

observables or the self-splicings of 12-link observablesdiscussed above.

We now direct our consideration back to the full set of_L-closed link strings of lengths 16 to 24. By exhaustive

exploration, we were able to show that all of them can beobtained as splicings of two shorter L-closed link strings.An explicit specification of the splicings is given in the listsof TDI observables available at the webpage www.vallis.org/tdi. A good portion of the _L-closed link strings

-13

FIG. 6. The time-reversal plus bridge symmetry is exploitedby certain _L-closed non-self-splicings at length 20 and above,such as the link string of Eq. (37).


are obtained from self-splicings. The non-self-splicingsoccur between L-closed strings that are often related byevident symmetries (such as index shifts). Interestingly,these symmetries may occur between L-closed strings ofdifferent length, as it happens for the 20-link observable

303�30��!

j1220 ��

1!

3031 ��

20213�220��!

330202 ��

; (37)

which can be interpreted as the splicing of the 8-link X

string 303220��!

330202 ��

with the reversal of the 12-link string

3031220 ��

1330��!

1

2021��!

. The latter can be obtained from theformer by reversing the time direction of all 303 and 220

bigrams and inserting 1!

and 1

bridges as needed to keepthe arrows consecutive, as shown in Fig. 6.

Also interesting is that some splicings result in _L-closedstrings shorter than the sum of the lengths of their compo-nents, as in the case of the 22-link observable

322021�10132��!

10123 ��

� 3!

2023 ��

1020��!

3122010 ��

32202110132��!

1012 ��

2023 ��

1020��!

3122010 ��

; (38)

which can be interpreted as the splicing of the 16-link

string 3220213��!

2023 ��

1020��!

3122010 ��

with the 8-link U-type

string 10132��!

10123 ��

, occurring in such a way that the final3

of the latter is erased by the subsequent 3!

from theformer [both bold in Eq. (38)].

We conjecture that all _L-closed strings can be obtainedas splicings of L-closed strings. A possible proof couldfocus on the monadic L-closed strings (those that cannot beobtained as the splicing of two L-closed strings), whichappear at every even length: consider for instance the string

303220220 � � � 220��!

330202202 � � � 202 ��

. To prove our conjec-ture, one would show that monadic L-closed strings arenever _L closed, as we know to be the case up to length 24.

IV. CONCLUSIONS

I have described Geometric TDI, a powerful new ap-proach to understand TDI intuitively, and to interpret all ofits observables as the measurements of virtual synthesizedinterferometers, extending the original intuition of Tintoand Armstrong [4] and the graphical explanations of

042003

Shaddock [10] and Summers [14]. In Geometric TDI,observables consisting of n one-way phase measurementsare represented by link strings of length n, and can beenumerated exhaustively by listing all possible link strings,and then applying simple rules to determine which stringscorrespond to first-generation (jLj-closed), modified(L-closed), or second-generation ( _L-closed) observables.A study of the closure symmetries of link strings providesclues to the general rule that modified TDI observablescombine (i.e., splice) into second-generation observables,maintaining in most cases the same GW sensitivity.

In addition to its pedagogical value, Geometric TDI hasthe practical interest of providing a systematic method toexplore the space of second-generation TDI observables;such a method was unavailable prior to this work. Possibleapplications include the optimization of GW sensitivity (inanalogy to the work of Prince and colleagues in Ref. [13])and the development of targeted noise diagnostics.

In Sec. III, I have used Geometric TDI to survey allsecond-generation TDI observables of lengths up to 24. Myresults (available in full at the webpage www.vallis.org/tdi)show that the garden of TDI observables contains a wealthof previously unknown specimens; this richness only in-creases with string length. In Sec. III B, I have pointed outhow certain alternative forms of the standard observables[such as the four-beam X1 observables of Eq. (22)] haveimproved GW sensitivity in realistic measurement condi-tions (because they have fewer noise and GW responsenulls) and reduced susceptibility to gaps and glitches (be-cause they have a smaller temporal footprint).

The new observables become possible in Geometric TDIbecause the time advancements and retardations of one-way phase measurements are put on the same footing,while only retardations were considered in traditionalTDI. This generalization does not constitute a third-generation TDI, but its combinatorial power and attractivesymmetry justify its addition to the canon of TDI. Thereseem to be no particular challenges to implementing time-symmetric observables, especially in the framework ofpost-processed TDI [25], whether performed onboard oron the ground.

Future work on Geometric TDI should endeavor toprove the two open conjectures formulated in this paper:,namely, that all second-generation TDI observables of anylength can be generated as splicings of two modified TDIobservables and that there are no second-generation TDIobservables of the perfect � type. (It also would be inter-esting to explore a notion of relaxed _L closure that includesthe �1, �2, and �3 variables described by Tinto and col-leagues [11].) Other promising directions of research in-clude the optimization of GW sensitivity (can it becharacterized geometrically?) and the exploration of gen-erative rules other than splicing for second-generation TDIobservables (what is the geometric counterpart and gener-alization of the algebra of observables studied inRefs. [7]?).

-14


ACKNOWLEDGMENTS

I wish to thank John Armstrong, Frank Estabrook, andMassimo Tinto for teaching me all things TDI; I thank alsoDaniel Shaddock and David Summers for useful discus-sions. I am grateful to the Supercomputing and Visuali-zation Facility at the Jet Propulsion Laboratory for provid-ing CPU time on their Orion cluster. This research wassupported by the LISA Mission Science Office at the JetPropulsion Laboratory, Caltech, where it was performedunder contract with the National Aeronautics and SpaceAdministration.

APPENDIX A: RULES AND PROOFS

1. Counting rule for _L-closed link strings

There is nothing magic about this rule, which is neces-sary and sufficient to have a null total light-travel time atfirst order/degree for generic arm lengths Ll�t�. To see why,set t � 0 at the beginning of the link string; steppingthrough the string, increment t by an advancement �l�t�(defined as the time experienced by light propagating alonglink l for emission at time t) for each symbol l

!, and

decrement t by a delay Ll�t� for each symbol l

. Eachadvancement �l or delay Ll enters the time arguments ofall subsequent �m’s and Lm’s in the string, generating termsf�l;�Llg � f _�m;� _Lmg in the first-order/degree Taylor ex-pansion of the total light-travel time. Since �l�t� � Ll�t��l�t�� ’ Ll�t� � Ll�t� _Ll�t�, we can replace all �l’s with Ll’s(while keeping the overall sum the same to first order/degree) by adding a term Ll _Ll for each l

!in the string.

The counting rule given in the main text is then equivalentto requiring that all the Ll _Lm terms cancel one by one toyield a null total light-travel time.

2. Translation rule from link strings to quasistandardTDI expressions

(1) S
tart at a bigram of type l!m (there must be at least
one in every closed loop), shifting the string cycli-cally to move the bigram close to the middle;

(2) m
ove to the left, starting with l!
, and write down ayij measurement for each index, according to thereplacement rules f1; 2; 3; 10; 20; 30g fy32; y13; y21;y23; y31; y12g; use a plus sign for! links and a minussign for links;

(3) w
hile doing this, build the delay sequence to beapplied to each new yij, adding a retardation ; r afterhaving translated each r!, and an advancement ; sbefore translating each s ;
(4) a
fter reaching the left end of the string, go back tothe link index m in the initial bigram and move tothe right, using the same replacement rules;
(5) w
hile doing this, build the delay sequence fromscratch, adding a retardation ; s after having trans-
042003-15

lated each s , and an advancement ; r before trans-lating each r!.

Shifting the string cyclically by a few positions can reducethe length of the delay sequence.

3. Proof that all self-splicings of L-closed observablesare _L closed

A few lemmas are needed for this proof. Looking backto the counting of l _m pairs outlined in Sec. II E, we denoteas prod[string] the polynomial obtained by regarding the36 possible l _m pairs as monomials, and sum them withcoefficients #�l _m

�!; l _m �� #� l

!_m ; l

_m!�. A string is _Lclosed if and only if prod�string� � 0.

Lemma 1.—The prod of an L-closed string is unchangedafter a cyclic string shift. Consider the shift of a singleindexm from one end of the string to the other. Because thestring is L closed, and must therefore have #� l

!� � #� l

�,

the contribution to prod[string] from the index m sumsdown to zero if m carries a ! , or to m _m if it carries a (and therefore does not multiply itself ). After the shift, thecontribution to prod[shifted string] from the index m is thesame. Hence this lemma.

Lemma 2.—The prods of an L-closed string and itsreversal are opposite numbers. This is established by notic-ing that prod�string� � prod�reversal� � �string� ��string� (i.e., the result of counting one l _m term, with theappropriate sign, for each l with each m in the string,including itself ). However, for a closed string, �string� ��string�must be zero; this is because in L-closed strings thecontribution from each index sums down to zero, given thateach index multiplies every other in equal numbers under! and . Hence this lemma.

Lemma 3.—The cross prod �string� � �reversal� of ashifted string and its shifted reversal (i.e., the result ofcounting one l _m term, with the appropriate sign, for eachl in the string and eachm in the reversal) is zero. In fact, thecross prod is separately zero for each index in the stringwith all the indices in its reversal, again because thereversal (as the original string) is L closed and has #�m!� �#�m �. Hence this lemma.

Lemma 4.—All self-splicings can be brought into anormal form given by the concatenation (j) of the shiftedstring and its shifted reversal, for appropriate shifts. Thisdoes not change the prod of the self-splicing.

Proof.—Hence, prod[self-splicing] is given by

prod�shifted stringjshifted reversal�

� prod�string� � prod�reversal�

� �shifted string� � �shifted reversal� � 0; (A1)

because the first two terms are opposite numbers and thethird term vanishes. Hence the proof.


4. Algebraic expressions for all second-generation TDIobservables of length 16

In this section I give explicit algebraic expressions for allthe second-generation TDI observables of length 16, asfound in my exhaustive survey, modulus the symmetriesdiscussed in Sec. II E. There is considerable arbitrariness inwriting these expressions, corresponding to the selection ofrepresentative link strings in each equivalence class, to theconvention used in translating strings to sums of yij mea-surements, and to the choice of the initial time of evalu-ation for each observable. Here I list link strings in anormal form whereby each string begins with the largestcontinuous substring of forward-time indices; I adopt thetranslation rules given in Sec. II E [just below Eq. (15)];and I adjust the time of evaluation to minimize the lengthof the longest delay sequence in the expression.

X type.—The standard 16-link observable X16;21 is ob-

tained by applying the translation rules to the link string

303220220303��!

202330330202 ��

, and evaluating the resulting ex-pression at the initial time t;303220220303:

y132;3220220303 � y231;220220303 � y123;20220303 � y321;220303

�y123;20303 � y321;303 � y132;3 � y231 � y3201 � y123;20

�y231;220 � y132;3220 � y231;303220 � y132;3303220

�y321;303303220 � y123;20303303220 ; (A2)

the alternative 16-link observables X16;4;�11 can be written

from the link strings 303220303��!

202330 ��

220��!

330202 ��

and

220303220��!

330202 ��

303��!

202330 ��

, evaluated at times t303220303202

and t220303220330

, respectively:

y132;3220303202

� y231;220303202

� y123;20303202

� y321;303202

�y132;3202

� y231;202

� y321;202

� y123;2 � y231

�y132;3 � y123;2303 � y321;202303 � y231;202303

�y132;3202303

� y321;303202303

� y123;20303202303

; (A3)

y123;20303220330

� y321;303220330

� y132;3220330

� y231;220330

�y123;20330

� y321;330

� y231;330

� y132;30

� y3201

�y123;20 � y132;30220 � y231;330220 � y321;330220

�y123;20330220

� y231;220330220

� y132;3220330220

; (A4)

last, the null 16-link observable X16;4;01 can be written from

the link string 220220��!

330202 ��

303303��!

202330 ��

, evaluated attime t

220220330202:

042003

y123;20220330202

� y321;220330202

� y123;20330202

� y321;330202

�y231;330202

� y132;30202

� y321;202

� y123;2 � y132;30

�y231;330

� y132;30330

� y231;330330

� y321;330330

�y123;20330330

� y231;220330330

� y132;3220330330

: (A5)

Expressions for the X-type observables that use the arms 1,2 and 1, 3 are obtained by cyclic index shifts.

U type.—The three 16-link U-type observables that usethe oriented arms 1, 10, 2, and 3 correspond to the linkstrings

32110��!

23123 ��

10132��!

10110 ��

; 32110��!

23 �

10132��!

10123110 ��

;

32110132��!

10123 ��

10!

23110 ��

;

(A6)

applying the translation rules and evaluating at the timest321102312, t3211023101, and t321101321, respectively, yields

y231;21102312 � y123;1102312 � y312;102312 � y213;2312

�y123;2312 � y231;312 � y312;12 � y123;2 � y231

�y213;103

� y312;1103

� y231;31103

� y123;231103

�y213;231103

� y312;10231103

� y213;110231103

; (A7)

y231;211023101 � y123;11023101 � y312;1023101 � y213;23101

�y123;23101 � y231;3101 � y213;1 � y312 � y231;3

�y123;23 � y213;23 � y312;1023 � y123;11023

�y231;211023 � y312;3211023 � y213;13211023; (A8)

y231;211013210

� y123;11013210

� y312;1013210

� y213;13210

�y312;3210

� y231;210

� y123;10

� y213;10

� y312

�y123;1 � y231;21 � y213;10321 � y123;10321

�y231;210321

� y312;3210321

� y213;13210321

; (A9)

the third expression is closest to the U1 given in Ref. [11](but see footnote 10). Expressions for the U-type observ-ables that other sets of oriented arms (i.e., f2; 20; 1; 3g,f3; 30; 1; 2g, f1; 10; 20; 30g, f2; 20; 10; 30g, and f3; 30; 10; 20g) areobtained by cyclic and noncylic index shifts.

E type.—The three 16-link E-type observables that usethe oriented arms 1, 10, 20, and 3 correspond to the linkstrings

11013��!

20101 ��

20!

3

1020��!

3110 ��

3!

20 ;

1011020��!

3110 ��

3!

20

13�!

20101 ��

20!

3 ;

11020��!

3

1013��!

20101 ��

20!

3110 ��

3!

20 ;

(A10)

applying the translation rules and evaluating at the times

-16


t110132010120

, t101102031103

, and t110203101320

, respectively, yields:

y312;10132010120

� y213;132010120

� y312;32010120

� y231;2010120

�y321;2010120

� y213;10120

� y312;120 � y3201 � y231

�y213;103

� y321;20103

� y231;20103

� y312;320103

�y213;1320103

� y231;3101320103

� y321;3101320103

; (A11)

y213;1102031103

� y312;102031103

� y213;2031103

� y321;31103

�y231;31103

� y312;1103

� y213;103

� y231 � y3201

�y312;120 � y231;3120 � y321;3120 � y213;203120

�y312;10203120 � y321;20110203120 � y231;20110203120 ; (A12)

y312;10203101320

� y213;203101320

� y321;3101320

� y231;3101320

�y213;1320

� y312;320

� y231;20

� y321;20

� y2103

�y312;10 � y321;20110 � y231;20110 � y312;320110

�y213;1320110

� y231;3101320110

� y321;3101320110

; (A13)

the third expression is closest to the E1 given in Ref. [11].Expressions for the other possible sets of oriented arms(i.e., f2; 20; 1; 30g and f3; 30; 10; 2g) are obtained by cyclicindex shifts.

P type.—The three 16-link P-type observables that usethe oriented arms 1, 10, 2, and 30 correspond to the linkstrings

3010110��!

2

30!

11030 ��

21�!

30

2!

1012 ��

;

21101��!

30

2!

1012 ��

3010��!

2

30!

11030 ��

;

2110��!

2

30101��!

30

2!

1012 ��

30!

11030 ��

;

(A14)

042003

applying the translation rules and evaluating at the timest3010110230110

, t21101302101

, and t211023010130

, respectively, yields

y132;10110230110

� y213;110230110

� y312;10230110

� y213;230110

�y123;230110

� y132;110

� y312;110

� y213;10

� y1302

�y123;230 � y312;1230 � y132;1230 � y123;2301230

�y213;2301230 � y312;102301230 � y123;1102301230 ; (A15)

y123;1101302101

� y312;101302101

� y213;1302101

� y312;302101

�y132;302101

� y123;101

� y213;101

� y312;1 � y123

�y132;302

� y213;10302

� y123;10302

� y132;30210302

�y312;30210302

� y213;130210302

� y132;10130210302

; (A16)

y123;11023010130

� y312;1023010130

� y213;23010130

� y123;23010130

�y132;10130

� y213;130

� y312;30

� y132;30

� y123;2

�y213;2 � y312;102 � y123;1102 � y132;3021102

�y312;3021102

� y213;13021102

� y132;1013021102

; (A17)

the third expression is closest to the P1 given in Ref. [11].Expressions for the other possible sets of oriented arms(i.e., f2; 20; 10; 3g and f3; 30; 1; 20g) are obtained by cyclicindex shifts.

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Detection and Observation of Gravitational Waves, Pre-Phase A Report,’’ Max Planck Institut fur Quantenoptik,Garching, Germany, 1998, 2nd ed.

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[3] See, e.g., S. A. Hughes, Ann. Phys. (N.Y.) 303, 142 (2003)and references therein.

[4] M. Tinto and J. W. Armstrong, Phys. Rev. D 59, 102003(1999).

[5] J. W. Armstrong, F. B. Estabrook, and M. Tinto,Astrophys. J. 527, 814 (1999); J. W. Armstrong, F. B.Estabrook, and M. Tinto, Classical Quantum Gravity 18,4059 (2001); see also G. Giampieri et al., Opt. Commun.123, 669 (1996) for a frequency-domain laser-noise sub-traction scheme related to TDI.

[6] F. B. Estabrook, M. Tinto, and J. W. Armstrong, Phys. Rev.D 62, 042002 (2000).

[7] S. V. Dhurandhar, K. R. Nayak, and J.-Y. Vinet, Phys. Rev.D 65, 102002 (2002); K. R. Nayak and J.-Y. Vinet, Phys.Rev. D 70, 102003 (2004).

[8] D. A. Shaddock, Phys. Rev. D 69, 022001 (2004).[9] N. Cornish and R. W. Hellings, Classical Quantum Gravity

20, 4851 (2003).[10] D. A. Shaddock, M. Tinto, F. B. Estabrook, and J. W.

Armstrong, Phys. Rev. D 68, 061303(R) (2003).

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[11] M. Tinto, F. B. Estabrook, and J. W. Armstrong, Phys.Rev. D 69, 082001 (2004).

[12] M. Tinto, D. A. Shaddock, J. Sylvestre, and J. W.Armstrong, Phys. Rev. D 67, 122003 (2003).

[13] T. A. Prince, M. Tinto, S. L. Larson, and J. W. Armstrong,Phys. Rev. D 66, 122002 (2002).

[14] D. Summers, ‘‘Proceedings of the Third Progress Meetingof the ESA-funded LISA PMS Project, ESTEC, theNetherlands, 2003’’; D. Summers and D. Hoyland,Classical Quantum Gravity 22, S249 (2005).

[15] See, e.g., G. M. Jenkins and D. G. Watts, (Holden-Day,San Francisco, 1968).

[16] C. W. Misner, K. S. Thorne, and J. A. Wheeler, Gravitation(Freeman, San Francisco, 1973).

[17] J. W. Armstrong, F. B. Estabrook, and H. D. Wahlquist,Astrophys. J. 318, 536 (1987); B. Bertotti et al., Astron.Astrophys. 296, 13 (1995); M. Tinto, Classical QuantumGravity 19, 1767 (2002); J. W. Armstrong, L. Iess, P.Tortora, and B. Bertotti, Astrophys. J. 599, 806 (2003).

[18] A. N. Lommen and D. C. Backer, Astrophys. J. 562, 297(2001); F. A. Jenet, A. Lommen, S. L. Larson, and L. Wen,Astrophys. J. 606, 799 (2004).

[19] F. B. Estabrook and H. D. Wahlquist, Gen. Relativ. Gravit.6, 439 (1975); H. D. Wahlquist, Gen. Relativ. Gravit. 19,1101 (1987).

042003

[20] M. Vallisneri, Phys. Rev. D 71, 022001 (2005); see alsowww.vallis.org/syntheticlisa.

[21] N. Ashby, Living Rev. Relativity 6, 1 (2003),www.livingreviews.org/lrr-2003-1.

[22] M. Tinto, J. W. Armstrong, and F. B. Estabrook, Phys.Rev. D 63, 021101(R) (2001); C. J. Hogan and P. L.Bender, Phys. Rev. D 64, 062002 (2001).

[23] ‘‘I received a telephone call one day at the graduatecollege at Princeton from Professor Wheeler, in whichhe said, ‘Feynman, I know why all electrons have the samecharge and the same mass.’ ‘Why?’ ‘Because, they are allthe same electron!’ [. . .] I did not take the idea that all theelectrons were the same one from him as seriously as Itook the observation that positrons could simply berepresented as electrons going from the future to the pastin a back section of their world lines,’’ quoted from R. P.Feynman, ‘‘The Development of the Space-Time View ofQuantum Electrodynamics,’’ Nobel Lecture, Dec. 11,1965.

[24] M. Tinto, M. Vallisneri, and J. W. Armstrong, Phys. Rev. D71, 041101(R) (2005).

[25] D. A. Shaddock, B. Ware, R. E. Spero, and M. Vallisneri,Phys. Rev. D 70, 081101(R) (2004).

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geometric time delay interferometry

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